1. **Problem Statement:** Given the complex number $z = \cos t + i \sin t$, find the magnitude $|z - 1|$ and express it in terms of $t$. Also, analyze the differential $dz$ and its magnitude $|dz|$. 2. **Recall the formula for magnitude of a complex number:** For $z = x + iy$, the magnitude is $|z| = \sqrt{x^2 + y^2}$. 3. **Calculate $|z - 1|$:** $$z - 1 = (\cos t - 1) + i \sin t$$ So, $$|z - 1| = \sqrt{(\cos t - 1)^2 + (\sin t)^2}$$ 4. **Simplify the expression inside the square root:** $$ (\cos t - 1)^2 + \sin^2 t = \cos^2 t - 2 \cos t + 1 + \sin^2 t $$ Using the Pythagorean identity $\cos^2 t + \sin^2 t = 1$, this becomes: $$ 1 - 2 \cos t + 1 = 2 - 2 \cos t $$ 5. **Rewrite the magnitude:** $$|z - 1| = \sqrt{2 - 2 \cos t} = \sqrt{2} \sqrt{1 - \cos t}$$ 6. **Use the half-angle identity:** $$1 - \cos t = 2 \sin^2 \left(\frac{t}{2}\right)$$ Therefore, $$|z - 1| = \sqrt{2} \sqrt{2 \sin^2 \left(\frac{t}{2}\right)} = \sqrt{2} \cdot \sqrt{2} \cdot \sin \left(\frac{t}{2}\right) = 2 \sin \left(\frac{t}{2}\right)$$ 7. **Analyze $dz$ and $|dz|$:** Given $z = e^{i t}$, then $$dz = i e^{i t} dt$$ The magnitude is $$|dz| = |i e^{i t}| dt = 1 \cdot dt = dt$$ since $|i| = 1$ and $|e^{i t}| = 1$. **Final answers:** $$|z - 1| = 2 \sin \left(\frac{t}{2}\right)$$ $$|dz| = dt$$